Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 014, 43 pages      arXiv:1109.0080      https://doi.org/10.3842/SIGMA.2012.014
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Emergent Braided Matter of Quantum Geometry

Sundance Bilson-Thompson a, Jonathan Hackett b, Louis Kauffman c and Yidun Wan d
a) School of Chemistry and Physics, University of Adelaide, SA 5005, Australia
b) Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada
c) Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045, USA
d) Open Research Centre for Quantum Computing, Kinki University, Kowakae 3-4-1, Higashi-osaka 577-0852, Japan

Received August 31, 2011, in final form March 12, 2012; Published online March 24, 2012

Abstract
We review and present a few new results of the program of emergent matter as braid excitations of quantum geometry that is represented by braided ribbon networks. These networks are a generalisation of the spin networks proposed by Penrose and those in models of background independent quantum gravity theories, such as Loop Quantum Gravity and Spin Foam models. This program has been developed in two parallel but complimentary schemes, namely the trivalent and tetravalent schemes. The former studies the braids on trivalent braided ribbon networks, while the latter investigates the braids on tetravalent braided ribbon networks. Both schemes have been fruitful. The trivalent scheme has been quite successful at establishing a correspondence between braids and Standard Model particles, whereas the tetravalent scheme has naturally substantiated a rich, dynamical theory of interactions and propagation of braids, which is ruled by topological conservation laws. Some recent advances in the program indicate that the two schemes may converge to yield a fundamental theory of matter in quantum spacetime.

Key words: quantum gravity; loop quantum gravity; spin network; braided ribbon network; emergent matter; braid; standard model; particle physics; unification; braided tensor category; topological quantum computation.

pdf (2210 kb)   tex (2695 kb)

References

  1. Alesci E., Bianchi E., Rovelli C., LQG propagator. III. The new vertex, Classical Quantum Gravity 26 (2009), 215001, 8 pages, arXiv:0812.5018.
  2. Alesci E., Rovelli C., Complete LQG propagator: difficulties with the Barrett-Crane vertex, Phys. Rev. D 76 (2007), 104012, 22 pages, arXiv:0708.0883.
  3. Alesci E., Rovelli C., Complete LQG propagator. II. Asymptotic behavior of the vertex, Phys. Rev. D 77 (2008), 044024, 11 pages, arXiv:0711.1284.
  4. Ashtekar A., Lectures on nonperturbative canonical gravity, Advanced Series in Astrophysics and Cosmology, Vol. 6, World Scientific Publishing Co. Inc., River Edge, NJ, 1991.
  5. Ashtekar A., Bombelli L., Corichi A., Semiclassical states for constrained systems, Phys. Rev. D 72 (2005), 025008, 16 pages, gr-qc/0504052.
  6. Baez J., Muniain J.P., Gauge fields, knots and gravity, Series on Knots and Everything, Vol. 4, World Scientific Publishing Co. Inc., River Edge, NJ, 1994.
  7. Baez J.C., An introduction to spin foam models of BF theory and quantum gravity, in Geometry and Quantum Physics (Schladming, 1999), Lecture Notes in Phys., Vol. 543, Springer, Berlin, 2000, 25-93, gr-qc/9905087.
  8. Baez J.C., Spin foam models, Classical Quantum Gravity 15 (1998), 1827-1858, gr-qc/9709052.
  9. Bahr B., Thiemann T., Gauge-invariant coherent states for loop quantum gravity. I. Abelian gauge groups, Classical Quantum Gravity 26 (2009), 045011, 22 pages, arXiv:0709.4619.
  10. Barrett J.W., Crane L., Relativistic spin networks and quantum gravity, J. Math. Phys. 39 (1998), 3296-3302, gr-qc/9709028.
  11. Bianchi E., Dona' P., Speziale S., Polyhedra in loop quantum gravity, Phys. Rev. D 83 (2011), 044035, 17 pages, arXiv:1009.3402.
  12. Bilson-Thompson S., A topological model of composite preons, hep-ph/0503213.
  13. Bilson-Thompson S., Hackett J., Kauffman L., Particle identifications from symmetries of braided ribbon network invariants, arXiv:0804.0037.
  14. Bilson-Thompson S., Hackett J., Kauffman L.H., Particle topology, braids, and braided belts, J. Math. Phys. 50 (2009), 113505, 16 pages, arXiv:0903.1376.
  15. Bilson-Thompson S.O., Markopoulou F., Smolin L., Quantum gravity and the standard model, Classical Quantum Gravity 24 (2007), 3975-3993, hep-th/0603022.
  16. Borissov R., Major S., Smolin L., The geometry of quantum spin networks, Classical Quantum Gravity 13 (1996), 3183-3195, gr-qc/9512043.
  17. Boulatov D.V., A model of three-dimensional lattice gravity, Modern Phys. Lett. A 7 (1992), 1629-1646, hep-th/9202074.
  18. Brill D.R., Hartle J.B., Method of the self-consistent field in general relativity and its application to the gravitational geon, Phys. Rev. 135 (1964), B271-B278.
  19. Cherrington J.W., Recent developments in dual lattice algorithms, arXiv:0810.0546.
  20. Cherrington J.W., Christensen J.D., A dual non-Abelian Yang-Mills amplitude in four dimensions, Nuclear Phys. B 813 (2009), 370-382, arXiv:0808.3624.
  21. Christensen J.D., Livine E.R., Speziale S., Numerical evidence of regularized correlations in spin foam gravity, Phys. Lett. B 670 (2009), 403-406, arXiv:0710.0617.
  22. Crane L., 2-d physics and 3-d topology, Comm. Math. Phys. 135 (1991), 615-640.
  23. Crane L., Frenkel I.B., Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994), 5136-5154, hep-th/9405183.
  24. De Pietri R., Petronio C., Feynman diagrams of generalized matrix models and the associated manifolds in dimension four, J. Math. Phys. 41 (2000), 6671-6688, gr-qc/0004045.
  25. De Pietri R., Rovelli C., Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity, Phys. Rev. D 54 (1996), 2664-2690, gr-qc/9602023.
  26. Dittrich B., Thiemann T., Are the spectra of geometrical operators in loop quantum gravity really discrete?, J. Math. Phys. 50 (2009), 012503, 11 pages, arXiv:0708.1721.
  27. Dowker H.F., Sorkin R.D., A spin-statistics theorem for certain topological geons, Classical Quantum Gravity 15 (1998), 1153-1167, gr-qc/9609064.
  28. Dreyer O., Markopoulou F., Smolin L., Symmetry and entropy of black hole horizons, Nuclear Phys. B 744 (2006), 1-13, hep-th/0409056.
  29. Ehrenfest P., In what way does it become manifest in the fundamental laws of physics that space has three dimensions?, Proc. Royal Netherlands Acad. Arts Sci. 20 (1917), 200-209.
  30. Finkelstein D., Misner C.W., Some new conservation laws, Ann. Physics 6 (1959), 230-243.
  31. Foxon T.J., Spin networks, Turaev-Viro theory and the loop representation, Classical Quantum Gravity 12 (1995), 951-964, gr-qc/9408013.
  32. Freidel L., Group field theory: an overview, Internat. J. Theoret. Phys. 44 (2005), 1769-1783, hep-th/0505016.
  33. Freidel L., Krasnov K., A new spin foam model for 4D gravity, Classical Quantum Gravity 25 (2008), 125018, 36 pages, arXiv:0708.1595.
  34. Furmanski W., Kolawa A., Yang-Mills vacuum: an attempt at lattice loop calculus, Nuclear Phys. B 291 (1987), 594-628.
  35. Gu Z.C., Levin M., Swingle B., Wen X.G., Tensor-product representations for string-net condensed states, Phys. Rev. B 79 (2009), 085118, 10 pages, arXiv:0809.2821.
  36. Gu Z.C., Levin M., Wen X.G., Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions, Phys. Rev. B 78 (2008), 205116, 11 pages, arXiv:0807.2010.
  37. Gu Z.C., Wen X.G., A lattice bosonic model as a quantum theory of gravity, gr-qc/0606100.
  38. Hackett J., Invariants of braided ribbon networks, arXiv:1106.5096.
  39. Hackett J., Invariants of spin networks from braided ribbon, arXiv:1106.5095.
  40. Hackett J., Locality and translations in braided ribbon networks, Classical Quantum Gravity 24 (2007), 5757-5766, hep-th/0702198.
  41. Hackett J., Wan Y., Conserved quantities for interacting 4-valent braids in quantum gravity, Classical Quantum Gravity 26 (2009), 125008, 14 pages, arXiv:0803.3203.
  42. Hackett J., Wan Y., Infinite degeneracy of states in quantum gravity, J. Phys. Conf. Ser. 306 (2011), 012053, 7 pages, arXiv:0811.2161.
  43. Hamma A., Markopoulou F., Premont-Schwarz I., Severini S., Lieb-Robinson bounds and the speed of light from topological order, Phys. Rev. Lett. 102 (2009), 017204, 4 pages, arXiv:0808.2495.
  44. Harari H., A schematic model of quarks and leptons, Phys. Lett. B 88 (1979), 83-86.
  45. He S., Wan Y., C, P, and T of braid excitations in quantum gravity, Nuclear Phys. B 805 (2008), 1-23, arXiv:0805.1265.
  46. He S., Wan Y., Conserved quantities and the algebra of braid excitations in quantum gravity, Nuclear Phys. B 804 (2008), 286-306, arXiv:0805.0453.
  47. Holbrook J.A., Kribs D.W., Laflamme R., Noiseless subsystems and the structure of the commutant in quantum error correction, Quantum Inf. Process. 2 (2003), 381-419, quant-ph/0402056.
  48. Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
  49. Kauffman L.H., Map coloring, q-deformed spin networks, and Turaev-Viro invariants for 3-manifolds, Internat. J. Modern Phys. B 6 (1992), 1765-1794.
  50. Kauffman L.H., Lomonaco Jr. S.J., q-deformed spin networks, knot polynomials and anyonic topological quantum computation, J. Knot Theory Ramifications 16 (2007), 267-332, quant-ph/0606114.
  51. Kempe J., Bacon D., Lidar D.A., Whaley K.B., Theory of decoherence-free fault-tolerant universal quantum computation, Phys. Rev. A 63 (2001), 042307, 29 pages, quant-ph/0004064.
  52. Kogut J., Susskind L., Hamiltonian formulation of Wilson's lattice gauge theories, Phys. Rev. D 11 (1975), 395-408.
  53. Konopka T., Markopoulou F., Severini S., Quantum graphity: a model of emergent locality, Phys. Rev. D 77 (2008), 104029, 15 pages, arXiv:0801.0861.
  54. Konopka T., Markopoulou F., Smolin L., Quantum graphity, hep-th/0611197.
  55. Kribs D., Laflamme R., Poulin D., Unified and generalized approach to quantum error correction, Phys. Rev. Lett. 94 (2005), 180501, 4 pages, quant-ph/0412076.
  56. Kribs D.W., Markopoulou F., Geometry from quantum particles, gr-qc/0510052.
  57. Kuratowski K., Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930), 271-283.
  58. Levin M., Wen X.G., Photons and electrons as emergent phenomena, Rev. Modern Phys. 77 (2005), 871-879, cond-mat/0407140.
  59. Levin M., Wen X.G., String-net condensation: a physical mechanism for topological phases, Phys. Rev. B 71 (2005), 045110, 21 pages, cond-mat/0404617.
  60. Lieb E.H., Robinson D.W., The finite group velocity of quantum spin systems, Comm. Math. Phys. 28 (1972), 251-257.
  61. Loll R., Volume operator in discretized quantum gravity, Phys. Rev. Lett. 75 (1995), 3048-3051, gr-qc/9506014.
  62. Major S., Q-quantum gravity, Ph.D. thesis, The Pennsylvania State University, 1997.
  63. Major S., Smolin L., Quantum deformation of quantum gravity, Nuclear Phys. B 473 (1996), 267-290, gr-qc/9512020.
  64. Markopoulou F., Conserved quantities in background independent theories, J. Phys. Conf. Ser. 67 (2007), 012019, 11 pages, gr-qc/0703027.
  65. Markopoulou F., Dual formulation of spin network evolution, gr-qc/9704013.
  66. Markopoulou F., Poulin D., Noiseless subsystems and the low energy limit of spin foam models, unpublished.
  67. Markopoulou F., Prémont-Schwarz I., Conserved topological defects in non-embedded graphs in quantum gravity, Classical Quantum Gravity 25 (2008), 205015, 29 pages, arXiv:0805.3175.
  68. Markopoulou F., Smolin L., Quantum theory from quantum gravity, Phys. Rev. D 70 (2004), 124029, 10 pages, gr-qc/0311059.
  69. Moussouris J.P., Quantum models as spacetime based on recoupling theory, Ph.D. thesis, Oxford University, 1983.
  70. Ooguri H., Topological lattice models in four dimensions, Modern Phys. Lett. A 7 (1992), 2799-2810, hep-th/9205090.
  71. Oriti D., Quantum gravity as a quantum field theory of simplicial geometry, in Quantum Gravity, Birkhäuser, Basel, 2007, 101-126, gr-qc/0512103.
  72. Oriti D., Spin foam models of quantum spacetime, Ph.D. thesis, University of Cambridge, 2003, gr-qc/0311066.
  73. Oriti D., The group field theory approach to quantum gravity, in Approaches to Quantum Gravity - toward a New Understanding of Space, Time, and Matter, Cambridge University Press, Cambridge, 2009, 310-331, gr-qc/0607032.
  74. Pachner U., Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten, Abh. Math. Sem. Univ. Hamburg 57 (1987), 69-86.
  75. Penrose R., Angular monentum: an approach to combinatorial spacetime, Cambridge University Press, Cambridge, 1971.
  76. Penrose R., On the nature of quantum geometry, Freeman, San Francisco, 1972.
  77. Perinia C., Rovellia C., Speziale S., Self-energy and vertex radiative corrections in LQG, Phys. Lett. B 682 (2009), 78-84, arXiv:0810.1714.
  78. Perry G.P., Cooperstock F.I., Stability of gravitational and electromagnetic geons, Classical Quantum Gravity 16 (1999), 1889-1916, gr-qc/9810045.
  79. Reidemeister K., Knot theory, Chelsea, New York, 1948.
  80. Rovelli C., Comment on "Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?" by B. Dittrich and T. Thiemann, arXiv:0708.2481.
  81. Rovelli C., Loop quantum gravity, Living Rev. Relativ. 1 (1998), 1, 68 pages, gr-qc/9710008.
  82. Rovelli C., Loop quantum gravity, Living Rev. Relativ. 11 (2008), 5, 69 pages.
  83. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  84. Rovelli C., Smolin L., Discreteness of area and volume in quantum gravity, Nuclear Phys. B 442 (1995), 593-619, gr-qc/9411005.
  85. Rovelli C., Smolin L., Spin networks and quantum gravity, Phys. Rev. D 52 (1995), 5743-5759, gr-qc/9505006.
  86. Rowell E.C., Two paradigms for topological quantum computation, in Advances in Quantum Computation, Contemp. Math., Vol. 482, Amer. Math. Soc., Providence, RI, 2009, 165-177, arXiv:0803.1258.
  87. Sahlmann H., Thiemann T., Towards the QFT on curved spacetime limit of QGR. I. A general scheme, Classical Quantum Gravity 23 (2006), 867-908, gr-qc/0207030.
  88. Shupe M.A., A composite model of leptons and quarks, Phys. Lett. B 86 (1979), 87-92.
  89. Smolin L., An invitation to loop quantum gravity, in Quantum Theory and Symmetries, World Scientific Publishing Co., Singapore, 2004, 655-682, hep-th/0408048.
  90. Smolin L., Generic predictions of quantum theories of gravity, in Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, Cambridge University Press, Cambridge, 2009, 548-570, hep-th/0605052.
  91. Smolin L., Quantum gravity with a positive cosmological constant, hep-th/0209079.
  92. Smolin L., The case for background independence, in The Structural Foundations of Quantum Gravity, Oxford Univ. Press, Oxford, 2006, 196-239, hep-th/0507235.
  93. Smolin L., Wan Y., Propagation and interaction of chiral states in quantum gravity, Nuclear Phys. B 796 (2008), 331-359, arXiv:0710.1548.
  94. Thiemann T., Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity, Phys. Lett. B 380 (1996), 257-264, gr-qc/9606088.
  95. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
  96. Thiemann T., Quantum spin dynamics (QSD), Classical Quantum Gravity 15 (1998), 839-873, gr-qc/9606089.
  97. Thiemann T., Quantum spin dynamics (QSD). II. The kernel of the Wheeler-DeWitt constraint operator, Classical Quantum Gravity 15 (1998), 875-905, gr-qc/9606090.
  98. Thompson (Lord Kelvin) W., Hydrodynamies, Proc. Roy. Soc. 41 (1867), 94-105.
  99. Turaev V.G., Viro O.Y., State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31 (1992), 865-902.
  100. Wan Y., 2D Ising model with non-local links - a study of non-locality, hep-th/0512210.
  101. Wan Y., Effective theory of braid excitations of quantum geometry in terms of Feynman diagrams, Nuclear Phys. B 814 (2009), 1-20, arXiv:0809.4464.
  102. Wan Y., Emergent matter of quantum geometry, Ph.D. thesis, University of Waterloo, 2009.
  103. Wan Y., On braid excitations in quantum gravity, arXiv:0710.1312.
  104. Wang Z., Topologization of electron liquids with Chern-Simons theory and quantum computation, in Differential Geometry and Physics, Nankai Tracts Math., Vol. 10, World Sci. Publ., Hackensack, NJ, 2006, 106-120, cond-mat/0601285.
  105. Wheeler J.A., Geons, Phys. Rev. 97 (1955), 511-536.
  106. Zanardi P., Rasetti M., Noiseless quantum codes, Phys. Rev. Lett. 79 (1997), 3306-3309, quant-ph/9705044.


Previous article  Next article   Contents of Volume 8 (2012)